Band Theory of Solids

AUTONOMOUS

INTRODUCTION

• Bloch stated this theory in 1928. According to this theory, the free electrons moves in a periodic field provided by the lattice. This theory is also called Band theory of solids.

• The energy band theory of solids is the basic principle of semiconductor physics and it is used to explain the differences in electrical properties between metals, insulators and semiconductors.

Electron in a periodic potential – Bloch theorem

• A crystalline solid consists of a lattice which is composed of a large number of positive ion cores at regular intervals and the conduction electrons move freely throughout the lattice.

• The variation of potential inside the metallic crystal with the periodicity of the lattice is explained by Bloch theorem.

+ + + + ++ +

+ + + + ++ +

+ + + + ++ +

+ + + + ++ +

+ + + + ++ +

• The potential of the solid varies periodically with the periodicity of space lattice and the potential energy of the particle is zero near the nucleus of the +ve ion in the lattice and maximum when it is half way between the adjacent nuclei which are separated by interatomic spacing distance ‘a’.

X

V

V

One dimensional periodic potential in crystal.

Periodic positive ion cores Inside metallic crystals.

+ + + + ++ +

+ + + + ++ +

+ + + + ++ +

+ + + + ++ +

+ + + + ++ +

Bloch’s Theorem• Bloch’s Theorem states that for a

particle moving in the periodic potential, the wavefunctions ψ(x) are of the form

• uk(x) is a periodic function with the periodicity of the potential– The exact form depends on the potential

associated with atoms (ions) that form the solid

)()(

function periodic a is )( ,)()(

axuxu

xuwhereexux

kk

kikx

k

• The one dimensional Schrödinger equation

• The periodic potential V(x) may be defined by means of the lattice constant a as V(x)=V(x+a)

0][8

2

2

2

2

VE

h

m

dx

d1

0)]([8

2

2

2

2

axVE

h

m

dx

d

Bloch has shown that the one dimensional solution of the Schrödinger equation is

)()exp()(

3

)()exp()(

rUikrr

DIn

xUikxx

kK

kk

lattice. crystal a ofy periodicit with periodic a is (x) UWhere k

2

I

• Let us consider a linear chain of atoms of length L in one dimensional case with N number of atoms in the chain

• This is refered to as Bloch condition.Similarly, the complex conjugate of eq(4)

)4.().........exp()()(

)exp()()exp()(

)(exp{)()(

)3....().........()(

ikNaxNax

ikxxUikNaNax

NaxikNaxUNax

NaxUxU

kk

kk

kk

kk

)()()()(

)23()4(

)5).......(exp().()(

**

*

xxNaxNax

andFromEq

ikNaxNax

kkkk

kk

• This means that the electron is localized around any particular atom and the probability finding the electron the electron is same throughout the crystal .

• Behaviour of an electron in a periodic potential:(The Kronig-Penny Model):

• This model treats the potential found in actual crystal to the point of getting an exact solution of the Schrödinger equation. It assumes that the potential energy of an electron in a linear array of positive nuclei has the form of a periodic array of square wells as shown in fig.

X=0 X=a

X=b

Potential barrier between the atoms.

We will eventually letV and b 0 in the problem.

The Kronig-Penney Model

U2(

x) U1(x)

x

V

• The potential energy is equal to zero in the regions 0<x<a, and in the potential V0 in the regions - b<x<0.Each of the potential energy wells may be considered..

• The wave functions associated with this model can be calculated by solving Schrödinger equations for the two regions:

2........00)(2

1..............002

022

2

22

2

xbforVEm

dx

d

axforEm

dx

d

• Let us define real quantities α and β by

• Now ,since the wave function must have Bloch form ,we may expect that

• Substituting eq (4) in eq(2) we get the following the equation for uk(x)

3......).........(;)(22

0202

22 VE

EVmand

mE

4.).........()( xUex kikx

axforukdx

duik

dx

ud 00)(2 1

22121

2

5

00)(2 22222

2

xbforkdx

duik

dx

ud

0

0)()(

2

)()(1

xbforDeCeu

axforBeAeuxikxik

xkixKi

6

The soln of these equations may be written as

7

Where A,B,C,D are the constants .These solutions must be subjected to the Following boundary condition

bxaxbxax

xxxx

dx

du

dx

duuu

dx

du

dx

duuu

212

0

2

0

10201

;)()(

;)()(

1

8

• The first two condition are imposed because of the requirement of continuity of the wave function Ψ and its derivative dΨ/dx at x=0,and hence of u and du/dx;the remaining two conditions are required because of the periodicity of uk(x).

• The application of these boundary condition to eq(7) leads to the following four linear homogenous equations involving the constants A,B,C,D:

• A+B=C+D

bikbikakiak DeCeBeAe

ikDikCkBikAi)()()()(

),()()()(

9

The coefficient A,B,C,D can be determined by solving these equation s,and Wave functions calculated.this leads to the following equations;

)(coscoscoshsinsinh2

22

baKabab

10

• This equation quite complicated ,.Kronig and Penny considered the possibility that Vo tends to infinity and b approaches zero in such a way that the product Vob remains finite .

• The quantity lim(Vob) representing the barrier strength.

• In this possibility , the equation (10) becomes

kaaaSinbmV

coscos2

0

If we define the quantity P by

20

bamV

p

11

• Eq (11) reduces to

Kaa

ap coscos

sin

12

This is the condition for the solutions of the wave equation to exist.

We see that this is satisfied only for those values of αa for which its Left hand side lies between +1and -1;this is because its right hand sideMust fall in this range .such values are represent the wave like solutions and are allowed.

•Consequence of this equationcan be understood with fig.

π

The Kronig-Penney Model

a

)cos()sin(

aa

aP

1

-1

Regions where the equation is satisfied, hence wherethe solution exists.

In general, as the energy increases (a increases), each successive band gets wider, and each successive gap gets narrower.

Boundaries are for αa = n.

No solutionexists, k2 < 0

0 2π 3π-π

-π

-2π

• The part of the vertical axis lying between the horizontal lines represents the range acceptable to the left-hand side

aa

ap

cossin

• Conclusions:

• **Allowed ranges of αa which permits a wave mechanical solution to exist are shown by the shadow portions. thus the motion of electrons in a periodic lattice is characterized by the bands of allowed energy separated by forbidden regions .

• ** As the value of α increase the width of the allowed energy bands also increase and the width of the forbidden band decreases.

• ** if the potential barrier strength P is large ,the function described by the right hand side of the equation crosses +1 and -1 region at steeper angle. Thus the allowed bands become narrower and forbidden bands become wider .

• If P tends to infinite the allowed band reduces to one single energy level :

a0

p

0p

a

If P tends to zero no energy levels exist, all energies are allowed to the electrons.

22

2

22

2

2

22

2

22

222

22

2

1

2)

2(

1)

2(

)2

)(8

(

)2

(

2

coscos

mvm

p

h

p

m

hE

m

hE

m

hE

km

E

mEk

k

k

kaa

Brillouin zones (E-k Curve)

• The Brillouin zone is a representation of permissive values of k of the electrons in one, two or three dimensions.

• Thus the energy spectrum of an electron moving in the presence of a periodic potential fields is divided into allowed zones and forbidden zones.

a

1

-1

d

d

2

d

3

d

4

d

d

2

d

3

d

4

The Kronig-Penney model gives us DETAILED solutions for the bands, which are almost, but not, cosinusoidal in nature.

Allowed bands

Energy gap

First Brillouin zone

E

k

Energy gap

a

a

2a

3

a

a

2

a

3

E-k diagram :

• When a parabola representing the energy of a free electron is compared with the energy of an electron in a periodic field.this parabola is discontinuities in the parabola occur at values of k given by

• k=nπ/a

Since k is the wave vector

k=2π/λ

nπ/a =2π/λ

2a=nλ

This is in the form of Bragg’s law .

I

• The solution to the wave equation under this condition yields two standing waves ,showing that two electron positions of differing potential energy are possible for the same value of k.This is gives rise to break in E-K curve.

• From the graph we find that the electron has allowed energy values in the region or zone extending from k=-π/a to +π/a. this zone is called first Brillouin zone

Origin of energy band formation in solids

• When we consider isolated atom, the electrons are tightly bound and have discrete, sharp energy levels.

• When two identical atoms are brought closer the outer most orbits of these atoms overlap and interact.

• If more atoms are brought together more levels are formed and for a solid of N atoms , each of the energy levels of an atom splits into N levels of energy.

• The levels are so close together that they form an almost continuous band.

• The width of this band depends on the degree of overlap of electrons of adjacent atoms and is largest for outer most atomic electrons.

N energy levels

N atoms

ΔE

E1

E2

E3

E2

E1

E1

• The energy bands in solids are important in determining many of physical properties of solids. The allowed energy bands (1) Valance band(2) Conduction band

• The band corresponding to the outer most orbit is called conduction band and the next inner band is called valence band. The gap between these two allowed bands is called forbidden energy gap.

Classifications of solids into Conductors, Semiconductors & Insulators

• On the basis of forbidden band or energy gap the solids are classified into insulators, semiconductors and conductors.Insulators:

• In case of insulators, the forbidden energy band is very wide as shown in figure.

• Due to this fact the electrons cannot jump from valance band to conduction band.

Forbidden gap

Valance band

Conduction band

INSULATORS

Forbidden gap

Valance band

Conduction band

SEMI CONDUCTORS

Valance band

Conduction band

CONDUCTORS

Semi conductors• In semi conductors the forbidden energy

( band ) gap is very small as shown in a figure.

• Ge and Si are the best examples of semiconductors.

• Forbidden ( band ) is of the order of 0.7ev & 1.1ev.

Conductors• In conductors there is no forbidden gap.

Valence and conduction bands overlap each other as shown in figure above.

• The electrons from valance band freely enter into conduction band.

Effective mass of an electron• The effective mass of an electron arises due to

periodic potential provided by the lattice.• When an electron in a periodic potential of lattice is

accelerated by an electric field, then the mass of the electron varies, mass is called effective mass of the electron m*.

• Consider an electron of charge e and mass m acted on by electric field.

Acceleration is not a constant in the periodic lattice of the crystal so mass of the electron replaced by its effective mass m* when it is moving in a periodic potential or crystal lattice.

m

eEa

eEma

eEf

*m

eEa

Consider the free electron as a wave packet moving with a velocity Vg

dk

dEv

dk

dE

hv

h

dEd

h

EhE

dk

dv

dk

dv

vectorwavek

frequencyangular

wheredk

dv

g

g

g

g

g

1

2

.,,

2

.

.2

Fdk

Eda

dt

dp

dk

Eda

dt

pd

dk

Eda

Fdt

dpand

pkcedt

dk

dk

Eda

dtdk

Eda

dt

dva g

2

2

2

2

2

2

2

2

2

2

2

1

)(1

))(

(1

..

.,sin

1

11

2

2

2

2

2

2

2

2

2

1

dkEd

m

dkEda

F

Fdk

Eda

The effective mass of an electron

}{2

2

2 dk

Edm

m

mfk

The degree of freedom of an electron is generally defined by a factor.

a. Variation of E with K

b. Variation of v with K

c.Variation of m* with K

d. Variation of fk with K

k

0

E

0

V

m

0a

a

0k

kf

)(a

)(b

)(c

)(d

• Variation of v with K:

k

0

E

0

V

m

0a

a

0k

kf

)(a

)(b

)(c

)(d

The variation of velocity with k fig (b).when k=0The velocity is zero and the value of k increases The velocity is increase reaching its maximum Value at k=k0 .k0corresponds to that point of Inflexion on E-k curve .beyond this inflexion pointThe velocity begins to decrease .finally assumes the Zero value at k=Π/a

• Variation of m* with K

k

0

E

0

V

m

0a

a

0k

kf

)(a

)(b

)(c

)(d

The variation of m*with k.near k=0 the effective mass Approaches m.as the value of k increase m*increase ,reaching its maximum value at the point of inflexion On the E-K curve .above the point of inflexion m*

Negative and as k tends to π/a,it decreases to small Negative value.

• Variation of fkwith k:

k

0

E

0

V

m

0a

a

0k

kf

)(a

)(b

)(c

)(d

The degree of freedom of an electron fk=m/m*

2

2

2 dk

Edmfk

Fkis measure of the extent to which an electron In state k is free.if m*is large ,fk is small i.e the particleBehaves as a ‘heavy’ particle .When fk=1 the electron behaves as a free electron .Note that fkis positive in the lower half of the band And negative in the upper half.